On Sep 27, 6:03 pm, Arturo Magidin <magi...@member.ams.org> wrote:|While Goedel was explicit in his model construction (from the|assumption that ZF is consistent, he takes a model for ZF and|constructs an "internal" model in which ZF+C holds), forcing works|differently; I'm not sure it would be accurate to say Cohen "found an|interpretation in which AC is false" in the same sense as Goedel's|work does.

Goedel's result does give an obvious interpretation(being true in his class L) which makes AC true.Cohen's proof is not so obvious about it, but couldbe construed as also giving an interpretation inwhich it is false.

The tricky thing about the operation in his proof isthat he adjoins a "generic" set to a specific model(the minimal model, assuming there is one), whichdoesn't give us an explicit such set. But therelation of "forcing" which he defines gives us aninterpretation of sentences (making them true ifthey are true for any such model, whatever thegeneric set was). It's less of an interpretation inthe sense that ~A being forced is not the same asA not being forced, while ~A being true in L is justthe same as A not being true in L.

There's a more recent style of forcing proof, whichuses what are called Boolean-valued models. Givena Boolean-valued model, and a sentence, the sentencehas a value which is some element of a Booleanalgebra (and can be other than 1=true or 0=false).Cohen's construction in effect supplies a value toeach sentence.

I've wondered now and then whether there was aconstruction that worked in the opposite direction.Despite what I wrote in another message, the wayAC was proved consistent was by trimming down theuniverse to something small enough to make AC true.Cohen then made AC false sometimes by adding to amodel in which it is true an element that breaks it.But our usual intuition is that AC is made true byallowing the universe to be big enough to includeall those choice sets and so on. I've wondered ifthere was a way to start with a universe in whichAC fails and somehow fluff it up into one in whichAC holds.

Cohen mentions in relation to his proof the issuethat in order to be able to add a set, in the waythat he does, one has to have a universe that doesn'talready contain all the possibilities. (What headds is in principle an actual set that happenednot to be in the minimal mode.) Perhaps one wantsto trim down the universe in which AC fails maybeto some equivalent countable universe first, ormaybe just define some idealized interpretationof set existence which makes AC true anyway.

My vague thought was to make the sets of the newuniverse be something like imagined maximum elementsof partially ordered sets satisfying the conditionsof Zorn's lemma. But I don't know if such aconstruction works out at all.